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Chng 2

i S Boole & Cc Cng Lun L

C01009 Digital Systems Chng 2 : i s Boole v Cc cng lun l

2

Ni dung

i s Boole

i s chuyn mch

Cc cng lun l


3

i s Boole

i s Boole c th gii bit n ln u tin bi

George Boole qua tc phm An Investigation of the

Laws of Thought vo nm 1854

i s Boole 2 phn t: cc hng v bin Boole ch c

mang 2 gi tr 0 hoc 1 ( LOW / HIGH )

Cc bin Boole biu din cho mt khong in p trn

ng dy hoc ti ng nhp/ng xut ca mch

Gi tr 0 hoc 1 c gi l mc lun l (logic level)

Mchlun l

ng nhp ng xut

A

x

F

y


4

i s Boole

i s Boole, cng tng t nh cc h i s khc,c xy dng thng qua vic xc nh ngha mt snhng vn c bn sau: Min (domain), l tp hp (set) cc phn t (element) m

trn nh ngha nn h i s

Tp hp cc php ton (operation) thc hin c trnmin

Mt tp hp cc nh (postulate), hay tin (axiom)c cng nhn khng qua chng minh. nh phi mbo tnh nht qun (consistency) v tnh c lp(independence)

Mt tp hp cc h qu (consequence) c gi l nh l(theorem), nh lut (law) hay quy tc (rule)


5

nh Huntington

Pht biu bi nh ton hc Anh E.V.Huntington trn

c s h thng ha cc cng trnh ca G. Boole

S dng cc php ton trong lun l mnh

(propositional logic)

1. Tnh ng (closure)

Tn ti min B vi t nht 2 phn t phn bit v 2 php

ton + v sao cho:

Nu x v y l cc phn t thuc B th x + y cng l 1

phn t thuc B (php cng lun l - logical addition)

Nu x v y l cc phn t thuc B th x y cng l 1

phn t thuc B (php nhn lun l - logical multiplication)


6

nh Huntington

2. Tnh ng nht (identity)Nu x l mt phn t trong min B th

Tn ti 1 phn t 0 trong B , gi l phn t ng nhtvi php ton + , tha mn tnh cht x + 0 = x

Tn ti 1 phn t 1 trong B , gi l phn t ng nhtvi php ton , tha mn tnh cht x 1 = x

3. Tnh giao hon (commutative)

Giao hon ca php + :x + y = y + x

Giao hon ca php :x y = y x


7

nh Huntington

4. Tnh phn phi (distributive)

Php c tnh phn phi trn php +

x (y + z) = (x y) + (x z)

Php + c tnh phn phi trn php

x + (y z) = (x + y) (x + z)

5. B (complementation)

Nu x l 1 phn t trong min B th s tn ti mt phn

t khc gi l x (hay x ), l phn t b ca x tha mn:

x + x = 1 v

x x = 0


8

Tnh cht i ngu (Duality)

Quan st cc nh Hungtinton, ta thy chng mang tnhi xng (symmetry) tc l cc nh xut hin theo cp

Mi nh trong 1 cp c th c xy dng t nh cnli bng cch Thay i cc php ton 2 ngi ( + | )

Thay i cc phn t ng nht ( 0 | 1 )

C th suy ra mt kt qu no t cc nh bng cch Hon i php ton + vi php ton

Hon i phn t ng nht 0 vi phn t ng nht 1

iu ny th hin tnh i ngu i s Boole


9

Cc nh l c bn (fundamental theorem)

Cc nh l c chng minh t cc nh Huntington v ccnh i ngu theo 2 cch Chng minh bng phn chng (contradiction) Chng minh bng quy np (induction)

nh l 1 (Null Law)

1.a x + 1 = 1 1.b x 0 = 0

nh l 2 (Involution)

(x ) = x

nh l 3 (Idempotency)

3.a x + x = x 3.b x x = x

nh l 4 (Absorption)

4.a x + x y = x 4.b x (x + y) = x


10

Cc nh l c bn

nh l 5 (Simplification)

5.a x + x y = x + y

5.b x (x + y ) = x y

nh l 6 (Associative Law)

6.a x + (y + z) = (x + y ) + z = x + y + z

6.b x (y z) = (x y) z = x y z

nh l 7 (Consensus)

7.a x y + x z + y z = x y + x z

7.b (x + y) (x + z) (y + z) = (x + y) (x + z)

nh l 8 (De Morgans Law)

8.a (x + y) = x y

8.b (x y) = x + y


11

Ti gin biu thc boole

Y = A(AB + ABC)

= A(AB(1 + C)) distributive

= A(AB(1)) Null Law

= A(AB) identity

= (AA)B Associative Law

= AB Idempotency


12

V d

Ti gin

x = ACD + ABCD

z = (A + B)(A+B)

De Morgans

z = ((a+c) . (b+d))


13

V d

Ti gin

x = ACD + ABCD

= CD( A + AB )

= CD( A + B ) = ACD + BCD

z = (A + B)(A+B)

= AA + AB + AB + BB = 0 + (A+A)B + B = B

De Morgans

z = ((a+c) . (b+d))

= (a+c) + (b+d) = ac + bd


14

Ti gin biu thc bool sau

a)

b)

Ti gin biu thc boole sau s dng nh l DeMorgan

Bi tp


15

i s chuyn mch (switching algebra)

i vi i s Boole, min khng b hn ch (khng c gii hnt ra i vi s lng cc phn t trong min)

Gii hn xem xt i s Boole vi 2 phn t ng nht.

i s Boole 2 phn t

Nm 1937, Claude Shannon hin thc i s Boole 2 phn tbng mch in vi cc chuyn mch (switch)

Chuyn mch l thit b c 2 v tr bn: tt (off) hay m (on)

2 v tr ny ph hp biu din cho 0 hay 1

i s Boole 2 phn t cn c gi l i s chuyn mch

Cc phn t ng nht c gi l cc hng chuyn mch (switchingconstant)

Cc bin (variable) biu din cc hng chuyn mch c gi l ccbin chuyn mch (switching variable) tn hiu


16

Bng s tht (Truth Table)

Phng tin m t s ph thuc ca ng xut vo mc lun l

(logic level) ti cc ng nhp ca mch

Lit k tt c cc t hp c th ca mc lun l ti cc ng nhp v

kt qu mc lun l tng ng ti ng xut ca mch

S t hp ca bng N-ng nhp: 2N

A B x

0 0 1

0 1 0

1 0 1

1 1 0

A B C x

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 0

1 0 0 0

1 0 1 0

1 1 0 0

1 1 1 1?

A

Bx


17

Cc php ton chuyn mch

i s chuyn mch s dng

cc php ton trong lun l

mnh vi tn gi khc

Php ton AND

Php ton 2 ngi tng

ng vi php nhn lun l

Php ton OR

Php ton 2 ngi tng

ng vi php cng lun l

x y x y x + y x

0 0 0 0 1

0 1 0 1 1

1 0 0 1 0

1 1 1 1 0

Php ton NOT Php ton 1 ngi tng

ng vi php b lun l

Bng s tht ccphp chuyn mch


18

Cc php ton chuyn mch

Cc php ton chuyn mch c th c hin thc bi mch phncng

Bng s tht c th s dng nh 1 cng c dng xc minhquan h gia cc php ton chuyn mch

S dng bng s tht chng minh nh l De Morgan(x + y) = x y

x y x y x + y (x + y) x y

0 0 1 1 0 1 1

0 1 1 0 1 0 0

1 0 0 1 1 0 0

1 1 0 0 1 0 0


19

Biu thc (expression) chuyn mch

Biu thc chuyn mch l mt quan h hu hn cc

hng, bin, biu thc chuyn mch lin kt vi nhau

bi cc php ton AND, OR v NOT

V d

y + 1 , x x + x , z ( x + y )

E = ( x + y z ) ( x + y ) + ( x + y )

literal c s dng m ch bin hay b ca bin


20

Biu thc (expression) chuyn mch...

Mt biu thc c th c chuyn thnh nhiu dng tng ngbng cch s dng cc lut Boole

E = (x + y z) (x + y) + (x + y)

E1 = x x + x y + x y z + y y z + x y E3 =x + x y

E2 = x + x (y + y z) + x y E4 =x + y

Ti sao phi chuyn i dng ca cc biu thc ?

Cc thnh phn tha (redundant) trong biu thc literal lp ( x x hay x + x) bin v b ( x x hay x + x) hng (0 hay 1)

Khng hin thc cc thnh phn tha ca biu thc vo mch


21

Hm (function) chuyn mch

Hm chuyn mch (switching function) l mt php gn xc nh

v duy nht ca nhng gi tr 0 v 1 cho tt c cc t hp gi tr

ca cc bin thnh phn

Hm c xc nh bi danh sch cc tr hm ti mi t hp gi

tr ca bin (bng s tht)

Tn ti nhiu biu thc biu din cho 1 hm

S lng hm chuyn mch vi n bin l 2 lu tha 2n

x y x y x y E1 = x + x y E2 = x + y

0 0 1 1 1 1 1

0 1 1 0 0 0 0

1 0 0 1 0 1 1

1 1 0 0 0 1 1


22

nh l khai trin Shannon

f(x1, x2, , xn) = x1 . f(1, x2, , xn)

+ x1 . f(0, x2, , xn)

f(x1, x2, , xn) = ( x1 + f(0, x2, , xn) )

. ( x1 + f(1, x2, , xn) )


23

Cc php ton chuyn mch khc

Php ton NAND

Php ton 2 ngi tng

ng vi (NOT AND)

Php ton NOR

Php ton 2 ngi tng

ng vi (NOT OR)

Php ton Exclusive OR

E = x y = x y + x y

Php ton XNOR (Ex. NOR) E = ( x y ) = x y + x y

Bin NAND NOR Ex. OR XNOR

x y (x . y) (x + y) x y (x y)

0 0 1 1 0 1

0 1 1 0 1 0

1 0 1 0 1 0

1 1 0 0 0 1


24

Cng lun l

i s chuyn mch c th thc hin cc cng vic

trong i tht, cn phi c

Thit b vt l thc hin cc php ton chuyn mch

Tn hiu vt l (in p, ) thay th cho cc bin chuyn

mch

Cng (gate) hay cng lun l (logic gate) l tn chung

dng gi cc thit b vt l thc hin cc php ton

chuyn mch vi chnh xc (accuracy) v thi gian

tr (delay) chp nhn c


25

Cng lun l

Mi cng c biu din bi 1 biu tng (schematic

symbol) c trng cng vi 1 s chn (pin, terminal)

tng trng cho cc bin chuyn mch

Mt biu thc chuyn mch bt k lun c th c

hin thc trong i tht bng cch kt ni cc cng

lun l li vi nhau

Mch lun l (logic circuit) hay mch chuyn mch

(switching circuit)


26

Biu tng ca cc cng lun l

Cng AND

Cng OR

Cng NOT (cng o - inverter)

Cng NAND

Cng NOR

Cng XOR

Cng XNOR

Cc cng nhiu hn 2 ngnhp

x

yx . y

x

yx + y

x

y(x . y)

x x

x

yx y

x

y(x y)

x

y(x + y)


27

Dng tng ng


28

Nguyn tc Bubble Pushing

y bong bng i ngc (t output) hoc i ti (t ngnhp) thay i tnh cht cng t AND sang OR vngc li.

y bong bng t ng ra sang ng nhp, bong bngxut hin trn tt c ng nhp, v tnh cht cng thayi.

y bong bng trn tt c ng nhp tin v ng xut.Bong bng xut hin trn ng xut v tnh cht cngthay i.

AB

YAB

Y

AB

YAB

Y


29

Din dch biu tng cng lun l

Dng tng ng ca cng AND

Ng xut mc cao khi tt c cc ng nhp mc cao

Ng xut mc thp khi mt trong cc ng nhp mc

thp

Mt s cu trc ca cng XOR

E = x y = x y + x y = ( x y + x y )


30

Tch cc cao Tch cc thp

Hai trng thi hot ng ca thit b l tch cc(activity) v khng tch cc (inactivity) Xt cc th d i vi in thoi, n, ng c, v.v

Do thi quen, qui c tch cc ng vi lun l 1 cnkhng tch cc ng vi lun l 0

Tch cc cao (active high)tch cc lun l 1 mc in p cao H

Tch cc thp (active low)tch cc lun l 0 mc in p thp L


31

Cng OR 7432

Cng NOR 7402

Cng Ex-OR 7486

Mch tch hp

Cng NOT 7404

Cng AND 7408

Cng NAND 7400


32

Tp ph bin ca cc php ton

Mt tp cc php ton c gi l ph bin (universal) nu mi

hm chuyn mch u c th c biu din mt cch tng

minh ch bi cc php ton ca tp trn

i vi cc php ton chuyn mch xt, ta c mt s cc tp

ph bin sau

Tp { NOT , AND , OR }

Tp { NOT , AND }

Tp { NOT , OR }

Tp { NAND }

Tp { NOR }

Tp ...

Bt k hm chuyn mch no cng u c th c biu din mt cch

tng minh ch bi cc php ton NOT v AND


33

Tnh ph bin ca cng NAND


34

Tnh ph bin ca cng NOR


35

Xc nh gi tr ng xut mch lun l

S dng biu thc Boole cho ng xut ca mch lun l

Vi A = 0, B = 1, C = 1, D = 1

x = AB C ( A + D )

= 0. 1 . 1 . (0 + 1)

= 1 . 1 . 1 . 1 = 1 . 1 . 1 . 0 = 0

S dng trc tip s mch lun l m khng cn s dng

biu thc Boolean


36

Gin xung theo thi gian (Timing Waveform)


37

Tng kt

i s Boolean 5 nh Huntington Tnh cht i ngu 8 nh l c bn

i s chuyn mch Thu gn i s Boolean cho min hai phn t {0,1} Cc php ton chuyn mch nh l khai trin Shannon

Cng lun l. Biu din cng lun l. Din dch cng lun l. Cc IC c bn. Tp ph bin php ton. Biu din s dng sng (Timing Waveform)


38

Tt c bi tp trong sch Digital System ca Ronal

Tocci

Chng 3 - Logic Gates and Boolean Algebra

Bi tp v nh v c thm


39

Bi tp c bn

a. V gin xung cho tn hiu ng ra X ca cng OR

b. Gi s tn hiu A trong hnh trn b ni tt vi t GND (A = 0). V gin

xung cho tn hiu X ca cng OR.

c. Gi s tn hiu A trong hnh trn b ni tt ln ngun +5V VCC (A = 1).

V gin xung cho tn hiu X ca cng OR.

d. Vi cng OR 5 ng nhp, c bao nhiu t hp ng nhp cho php ng xut

mc cao (HIGH or 1)?

A

B

C

1234C

BA

x


40

Bi tp c bn

Vit biu thc i s Boole v bng s tht cho ng

xut ca cc mch di y.

12

3 12

3X

12

3

12A

B

C

12

A

B

C

Y

12

3

12

3

12

3

12

C

B

A 12

3 Z

121

2

3

12

3


41

Bi tp c bn

Trnh by nguyn l hot ng ca h thng bo ng

di y, bit ci bo ng c kch hot khi tn hiu

iu khin mc cao (HIGH or 1)


42

Bi tp c bn

V cc mch lun l tng ng vi cc biu thc i s

Boole sau

= + + +

= ( + )

n gin cc biu thc sau:

= + + + + + +

+ + + +

= + + + + + ( + )


43

Bi tp c bn

n gin cc biu thc Boolean sau


45

Bi tp c bn

Ti gin cc biu thc sau

a. xyz + xyz + xy

b. (wx)(w+y)(xyz)

c. x(y+wyz) + xy(wz+z)

d. (w+x)(w+x+yz)(w+y)


46

Bi tp c bn

Chng minh bng i s cc biu thc sau


47

Bi tp c bn

Tm b ca cc biu thc sau y


48

Bi tp c bn

a) Xy dng 1 cng NAND 2 ng nhp ch s dng cc

cng NOR 2 ng nhp.

b) Xy dng 1 cng NOR 2 ng nhp ch s dng cc cng

NAND 2 ng nhp.

c) Hin thc biu thc = ch s dng 1 cng NOR

2 ng nhp v 1 cng NAND 2 ng nhp.

d) Hin thc biu thc = ch s dng cc cng

NAND 2 ng nhp.


49

Bi tp c bn

a. Bin i cc mch sau y ch s dng cng NAND

b. Bin i mch sau y ch s dng cng NOR

12

12

3

12

3

12

3

A

B

12

X

12

3 12

3X

12

3

12A

B

C

12

B

A

C

X1

2

3

12

3

12

12

3

A

B

C

Y

12

3

12

3

12

3

12


50

Bi tp c bn

V k hiu cng lun l thch hp cho cc pht biu sau

y:

Ng xut ch mc cao (HIGH or 1) khi c 3 ng nhp u

mc thp (LOW or 0).

Ng xut ch mc thp khi bt k ng nhp no trong 4

ng nhp mc thp.

Ng xut ch mc thp khi tt c 5 ng nhp u mc

cao.


51

Bi tp c bn

Cho s sau

Gi s ci bo ng c kch hot khi tn hiu iu khin

Z mc cao (HIGH or 1). Xc nh cc t hp ng nhp

tch cc h thng bo ng.

Gi s ci bo ng c kch hot khi tn hiu iu khin

Z mc thp (LOW or 0). Hy thay i s mch trn

phn nh r c ch hot ng ca h thng. T xc

nh cc t hp ng nhp tch cc h thng bo ng.


52

Bi tp c bn

Xc nh cc t hp ng nhp n LED sng

A

B

C

D

E

NOR

NAND

OR

R1

LED

+5V

NOT

NOT

NOT

AND


53

Bi tp m rng

Cho A.B = 0 v A + B = 1, chng minh ng thc sau:

A C + A B + B C = B + C

Cho hm F(A, B, C) c s logic nh hnh v.

a. Xc nh biu thc ca hm F(A, B, C)

b. Chng minh F c th thc hin ch bng 1 cng logic duy

nht.


54

Bi tp m rng

Chng minh cc ng thc sau bng i s boole.a. + + = + + +

b. + + = + + +

c. + + = + +

d. = B

e. () =


55

Bi tp m rng

Mt my bay phn lc p dng h thng kim sot tc quay

(rpm), p lc (pressure) v nhit (temperature) ca ng c s

dng cc sensor, vi chc nng nh sau:

RPM sensor xut 0 ch khi tc < 4800 rpm

P sensor xut 0 ch khi p sut < 220 psi

T sensor xut 0 ch khi nhit < 200F

Hnh bn di m t s hot ng ca n cnh bo. n cnh bo ch

sng khi W mc HIGH (=1)

a. Xc nh iu kin n cnh bo sng.

b. Thit k li mch ch s dng NAND.

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